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Fourier transforms of vector measures on some semigroups. (English) Zbl 0942.28019
The authors consider Fourier transforms of vector measures on the semi-group $$G$$ defined by a non-void set with a complete order relation. It is made into a semi-group by the operation $$xy=\max (x,y)$$. This semi-group is supposed to be compact.
Generalization of analogous results by the same authors are obtained.
The main result is the following: If $$X$$ is a Banach space and $$\Gamma$$ the semi-group of semicharacters of $$G$$, a function from $$\Gamma$$ into $$X$$ is the Fourier transform of regular $$X$$-valued Borel measure on $$G$$ if and only if it is weakly continuous and of bounded variation.
##### MSC:
 28E10 Fuzzy measure theory 28B05 Vector-valued set functions, measures and integrals 43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.